Finite Group Databasebeta

← C2×#87292 #11703600

Subgroups — C2×#87292 #11703600

Order 768.

11126 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×2−1+4C211yesmaximal, characteristic, complemented
2C22×#1010C211yesmaximal, characteristic, complemented
3#87292C2132yesmaximal, direct product factor
4C2×#5824C2112yesmaximal, complemented
5C2×#5825C213yesmaximal, complemented
6C2×#5832C213yesmaximal, complemented
7C2×Dic3×#41C211yesmaximal, characteristic, complemented
8C2×#5816C213yesmaximal, complemented
9C2×Q8×#67C214yesmaximal, complemented
10C22×#944C213yesmaximal, complemented
11C2×#3499—31nomaximal, Sylow
12C2×C3×2−1+4—216no—
13C2×C3×2−1+4C22116yescomplemented
14C22×C3×#41C2213yes—
15C22×C3×#41C2216yes—
16C22×C3×#41C2211yescharacteristic
17C23×C3×Q8C2213yes—
18C23×C3×Q8C2212yes—
19C2×#1010—216no—
20C2×#1010C22116yescomplemented
21C22×D3×Q8C2212yes—
22C22×#237C2216yes—
23C22×#235C2213yes—
24C23×Dic6C2213yes—
25#5832C22148yescomplemented
26Dic3×#41C22116yescomplemented
27#5825C22148yescomplemented
28#5824C221192yescomplemented
29#5816C22148yescomplemented
30Q8×#67C22164yescomplemented
31C2×#947C2213yes—
32C2×#946C22112yes—
33C2×Q8×Dic3C2212yes—
34C2×#944C2213yes—
35C2×#944C22112yes—
36C2×#944C2216yes—
37C2×#944C22148yescomplemented
38C2×#940C2211yescharacteristic
39C2×#939C22112yes—
40C2×#936C2213yes—
41C2×#935C22112yes—
42C2×D4×Dic3C2216yes—
43C2×#929C22112yes—
44C2×C4×#67C22112yes—
45C2×#926C2213yes—
46C2×#924C22112yes—
47C22×Dic3⋊C4C22112yes—
48C22×C4×Dic3C2213yes—
49C22×2−1+4—31no—
50C22×2−1+4D311yescharacteristic, complemented
51#3499—332no—
52C2×D4×Q8—34no—
53C2×#547—312no—
54C2×#536—33no—
55C2×#536—33no—
56C2×#532—33no—
57C22×C4⋊Q8—33no—
58C2×C4×#41—31no—
59C3×2−1+4—2128no—
60C2×C3×#41—248no—
61C2×C3×#41—224no—
62C2×C3×#41D4116yescomplemented
63C2×C3×#41C23148yes—
64C2×C3×#41C23124yes—
65C2×C3×#41C2318yes—
66C22×C3×Q8—216no—
67C22×C3×Q8—224no—
68C22×C3×Q8C2313yes—
69C22×C3×Q8C2316yes—
70C22×C3×Q8C23124yes—
71C22×C3×Q8C23116yes—
72C22×C3×Q8C2311yescharacteristic
73C22×C3×D4C2316yes—
74C22×C3×D4C2311yescharacteristic
75C22×C3×D4C2313yes—
76C23×C3×C4C2316yes—
77C23×C3×C4C2316yes—
78C23×C3×C4C2313yes—
79C22×#67C2316yes—
80C22×#67C2314yes—
81C23×Dic3C2316yes—
82C23×Dic3C2316yes—
83#1010—2128no—
84C2×D3×Q8—216no—
85C2×D3×Q8C23116yes—
86C2×#237—248no—
87C2×#237C23148yes—
88C2×#235—224no—
89C2×#235C23124yes—
90C22×D12C2311yescharacteristic
91C22×C4×D3C2316yes—
92C22×Dic6—224no—
93C22×Dic6C2313yes—
94C22×Dic6C2316yes—
95C22×Dic6C23124yes—
96C2×#217C23112yes—
97C2×#217C2316yes—
98#947C23124yes—
99#946C23196yes—
100Q8×Dic3C23116yes—
101#944C23124yes—
102#944C23196yes—
103#944C23148yes—
104#940C2318yes—
105#939C23196yes—
106#936C23124yes—
107#935C23196yes—
108D4×Dic3C23148yes—
109#929C23196yes—
110C4×#67C23196yes—
111C2×(C2×D3)⋊C4C23112yes—
112#926C23124yes—
113C2×C4⋊Dic3C2316yes—
114#924C23196yes—
115C2×Dic3⋊C4C23112yes—
116C2×Dic3⋊C4C23124yes—
117C2×Dic3⋊C4C23196yes—
118C2×C4×Dic3C2311yescharacteristic
119C2×C4×Dic3C2313yes—
120C2×C4×Dic3C2316yes—
121C2×C4×Dic3C23124yes—
122C2×2−1+4—216no—
123C2×2−1+4—316no—
124C2×2−1+4—616no—
125C2×2−1+4C2×D3116yescomplemented
126C22×#41—36no—
127C22×#41—33no—
128C22×#41C2×D313yes—
129C22×#41C2×D316yes—
130C22×#41C2×D311yescharacteristic
131C23×Q8—32no—
132C23×Q8—33no—
133C23×Q8C2×D313yes—
134C23×Q8C2×D312yes—
135D4×Q8—364no—
136#547—3192no—
137#536—348no—
138#536—348no—
139#532—348no—
140C2×C4⋊Q8—33no—
141C2×C4⋊Q8—36no—
142C2×C4⋊Q8—312no—
143C2×C4⋊Q8—348no—
144C2×#127—31no—
145C2×#124—33no—
146C2×#124—33no—
147C2×#123—312no—
148C2×#123—312no—
149C2×C22⋊Q8—312no—
150C2×C22⋊Q8—312no—
151C2×#121—312no—
152C4×#41—316no—
153C2×C4×Q8—32no—
154C2×C4×D4—36no—
155C2×C4×D4—312no—
156C2×#117—33no—
157C22×C4⋊C4—312no—
158C22×C42—33no—
159C24×C3C2412yes—
160C24×C3C2413yes—
161C23×D3C2412yes—
162C3×#41—2192no—
163C3×#41—296no—
164C3×#41C2×D4164yescomplemented
165C2×C3×Q8—264no—
166C2×C3×Q8—212no—
167C2×C3×Q8—224no—
168C2×C3×Q8—296no—
169C2×C3×Q8C2×D418yes—
170C2×C3×Q8C24112yes—
171C2×C3×Q8C24124yes—
172C2×C3×Q8C2414yes—
173C2×C3×D4—224no—
174C2×C3×D4—24no—
175C2×C3×D4C2×D4124yes—
176C2×C3×D4C24124yes—
177C2×C3×D4C2414yes—
178C2×C3×D4C24112yes—
179C22×C3×C4—224no—
180C22×C3×C4—224no—
181C22×C3×C4C2×D4124yes—
182C22×C3×C4C2416yes—
183C22×C3×C4C24124yes—
184C22×C3×C4C24124yes—
185C22×C3×C4C2411yescharacteristic
186C22×C3×C4C2413yes—
187C22×C3×C4C24112yes—
188C2×#67—224no—
189C2×#67—216no—
190C2×#67C24116yes—
191C2×#67C24124yes—
192C22×Dic3—224no—
193C22×Dic3—224no—
194C22×Dic3C2416yes—
195C22×Dic3C24124yes—
196C22×Dic3C2416yes—
197C22×Dic3C24124yes—
198C22×Dic3C2412yes—
199D3×Q8—264no—
200#237—2192no—
201#235—296no—
202C2×D12—24no—
203C2×D12C2414yes—
204C2×C4×D3—224no—
205C2×C4×D3C24124yes—
206C2×Dic6—212no—
207C2×Dic6—224no—
208C2×Dic6—296no—
209C2×Dic6C24124yes—
210C2×Dic6C24112yes—
211#217C24148yes—
212#217C24124yes—
213(C2×D3)⋊C4C24148yes—
214C4⋊Dic3C24124yes—
215Dic3⋊C4C24148yes—
216Dic3⋊C4C24196yes—
217C4×Dic3C2414yes—
218C4×Dic3C24112yes—
219C4×Dic3C24124yes—
2202−1+4—2128no—
2212−1+4—6128no—
222C2×#41—248no—
223C2×#41—224no—
224C2×#41—348no—
225C2×#41—324no—
226C2×#41—648no—
227C2×#41—624no—
228C2×#41#67116yescomplemented
229C2×#41C22×D3124yes—
230C2×#41C22×D3148yes—
231C2×#41C22×D318yes—
232C22×Q8—216no—
233C22×Q8—224no—
234C22×Q8—316no—
235C22×Q8—36no—
236C22×Q8—33no—
237C22×Q8—324no—
238C22×Q8—616no—
239C22×Q8—624no—
240C22×Q8C22×D316yes—
241C22×Q8C22×D313yes—
242C22×Q8C22×D3124yes—
243C22×Q8C22×D3116yes—
244C22×Q8C22×D311yescharacteristic
245C22×D4—34no—
246C22×D4—36no—
247C22×D4—31no—
248C22×D4C22×D316yes—
249C22×D4C22×D311yescharacteristic
250C22×D4C22×D313yes—
251C23×C4—36no—
252C23×C4—36no—
253C23×C4—36no—
254C23×C4C22×D316yes—
255C23×C4C22×D316yes—
256C23×C4C22×D313yes—
257C4⋊Q8—324no—
258C4⋊Q8—348no—
259C4⋊Q8—396no—
260#127—38no—
261#124—324no—
262#124—324no—
263#123—396no—
264#123—396no—
265C22⋊Q8—396no—
266C22⋊Q8—396no—
267#121—396no—
268C4×Q8—316no—
269C4×D4—348no—
270C4×D4—396no—
271#117—324no—
272C2×C4⋊C4—312no—
273C2×C4⋊C4—324no—
274C2×C4⋊C4—396no—
275C2×C4⋊C4—36no—
276C2×C22⋊C4—36no—
277C2×C22⋊C4—312no—
278C2×C22⋊C4—312no—
279C2×C42—33no—
280C2×C42—36no—
281C2×C42—324no—
282C2×C42—31no—
283C23×C3—212no—
284C23×C3—24no—
285C23×C3—28no—
286C23×C3C22×D4112yes—
287C23×C3C2514yes—
288C23×C3C2516yes—
289C23×C3C2511yescharacteristic
290C22×D3—24no—
291C22×D3—28no—
292C22×D3C2514yes—
293C3×Q8—224no—
294C3×Q8—248no—
295C3×Q8C22×D4116yes—
296C3×D4—248no—
297C3×D4—28no—
298C3×D4C22×D4148yes—
299C2×C3×C4—212no—
300C2×C3×C4—248no—
301C2×C3×C4—248no—
302C2×C3×C4C22×D414yes—
303C2×C3×C4C22×D4112yes—
304C2×C3×C4C22×D4148yes—
305C2×C3×C4C25112yes—
306C2×C3×C4C2512yes—
307C2×C3×C4C2516yes—
308#67—248no—
309#67—232no—
310C2×Dic3—212no—
311C2×Dic3—248no—
312C2×Dic3—212no—
313C2×Dic3—248no—
314C2×Dic32−1+418yescomplemented
315C2×Dic3C25112yes—
316C2×Dic3C25112yes—
317C2×Dic3C2514yes—
318D12—28no—
319C4×D3—248no—
320Dic6—224no—
321Dic6—248no—
322C24—32no—
323C24C23×D312yes—
324C24C23×D313yes—
325#41—2192no—
326#41—296no—
327#41—6192no—
328#41—696no—
329#41C2×#67164yescomplemented
330C2×Q8—264no—
331C2×Q8—224no—
332C2×Q8—212no—
333C2×Q8—296no—
334C2×Q8—324no—
335C2×Q8—312no—
336C2×Q8—664no—
337C2×Q8—624no—
338C2×Q8—612no—
339C2×Q8—696no—
340C2×Q8C2×#6718yes—
341C2×Q8C23×D3124yes—
342C2×Q8C23×D3112yes—
343C2×Q8C23×D314yes—
344C2×D4—224no—
345C2×D4—24no—
346C2×D4—316no—
347C2×D4—324no—
348C2×D4—34no—
349C2×D4—616no—
350C2×D4—624no—
351C2×D4—64no—
352C2×D4C2×#67124yes—
353C2×D4C23×D3124yes—
354C2×D4C23×D314yes—
355C2×D4C23×D3112yes—
356C22×C4—224no—
357C22×C4—224no—
358C22×C4—32no—
359C22×C4—324no—
360C22×C4—324no—
361C22×C4—36no—
362C22×C4—36no—
363C22×C4—324no—
364C22×C4—624no—
365C22×C4—624no—
366C22×C4—624no—
367C22×C4C2×#67124yes—
368C22×C4C23×D3124yes—
369C22×C4C23×D3124yes—
370C22×C4C23×D316yes—
371C22×C4C23×D3112yes—
372C22×C4C23×D311yescharacteristic
373C22×C4C23×D313yes—
374C4⋊C4—348no—
375C4⋊C4—396no—
376C4⋊C4—324no—
377C22⋊C4—324no—
378C22⋊C4—348no—
379C22⋊C4—348no—
380C42—34no—
381C42—312no—
382C42—324no—
383C22×C3—212no—
384C22×C3—224no—
385C22×C3—24no—
386C22×C3—28no—
387C22×C3—48no—
388C22×C3C23×D4112yes—
389C22×C3C23×D412yes—
390C22×C3C2×2−1+414yes—
391C22×C3C2611yescharacteristic
392C2×D3—24no—
393C2×D3—28no—
394C2×D3—48no—
395C3×C4—212no—
396C3×C4C23×D414yes—
397C3×C4C23×D4112yes—
398Dic3—212no—
399Dic3—212no—
400Dic3C2×2−1+418yescomplemented
401C23—212no—
402C23—28no—
403C23—24no—
404C23—34no—
405C23—68no—
406C23—64no—
407C23C22×#67112yes—
408C23C24×D314yes—
409C23C24×D316yes—
410C23C24×D311yescharacteristic
411Q8—248no—
412Q8—224no—
413Q8—648no—
414Q8—624no—
415Q8C22×#67116yes—
416D4—248no—
417D4—28no—
418D4—632no—
419D4—648no—
420D4—68no—
421D4C22×#67148yes—
422C2×C4—212no—
423C2×C4—248no—
424C2×C4—248no—
425C2×C4—38no—
426C2×C4—34no—
427C2×C4—312no—
428C2×C4—312no—
429C2×C4—648no—
430C2×C4—612no—
431C2×C4—612no—
432C2×C4—648no—
433C2×C4—648no—
434C2×C4C22×#67112yes—
435C2×C4C22×#6714yes—
436C2×C4C22×#67148yes—
437C2×C4C24×D3112yes—
438C2×C4C24×D312yes—
439C2×C4C24×D316yes—
440C2×C3—212no—
441C2×C3—44no—
442C2×C3#349914yescomplemented
443C2×C3C24×D411yescharacteristic
444C2×C3C22×2−1+412yes—
445D3—44no—
446C22—212no—
447C22—224no—
448C22—28no—
449C22—24no—
450C22—48no—
451C22—68no—
452C22—64no—
453C22—128no—
454C22D3×2−1+414yes—
455C22C23×#67112yes—
456C22C23×#6712yes—
457C22C25×D311yescharacteristic
458C4—212no—
459C4—38no—
460C4—612no—
461C4—612no—
462C4C23×#67112yes—
463C4C23×#6714yes—
464C3C2×#349911yescharacteristic, complemented, Sylow
465C2—212no—
466C2—44no—
467C2—124no—
468C2#8729214yesdirect product factor
469C2C2×D3×2−1+412yes—
470C2C24×#6711yescharacteristic
subgroupG/H
centerC23C24×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC22×C3×2−1+4C2
socleC23×C3C25
hypercenterC22×2−1+4D3
hypocenterC3C2×#3499
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×2−1+4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×C3×2−1+4C2semidirect
2. C22×#1010C2semidirect
3. #87292C2direct (N × H)
4. C2×#5824C2semidirect
5. C2×#5825C2semidirect
6. C2×#5832C2semidirect
7. C2×Dic3×#41C2semidirect
8. C2×#5816C2semidirect
9. C2×Q8×#67C2semidirect
10. C22×#944C2semidirect
13. C2×C3×2−1+4C22semidirect
20. C2×#1010C22semidirect
25. #5832C22semidirect
26. Dic3×#41C22semidirect
27. #5825C22semidirect
28. #5824C22semidirect
29. #5816C22semidirect
30. Q8×#67C22semidirect
37. C2×#944C22semidirect
50. C22×2−1+4D3semidirect
62. C2×C3×#41D4semidirect
125. C2×2−1+4C2×D3semidirect
164. C3×#41C2×D4semidirect
228. C2×#41#67semidirect
314. C2×Dic32−1+4semidirect
329. #41C2×#67semidirect
400. Dic3C2×2−1+4semidirect
442. C2×C3#3499semidirect
464. C3C2×#3499semidirect
468. C2#87292direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C3×2−1+42
C22×#10102
#872922
C2×#58242
C2×#58252
C2×#58322
C2×Dic3×#412
C2×#58162
C2×Q8×#672
C22×#9442
C2×#34993